#228185

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#3480	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,26,23,27,21,35)(19,28,30,24,31,39,36,40)(20,25,34,44,33,42,37,43)(29,41,32,38,45,48,46,47);(17,19,22,30,23,31,21,36)(18,24,26,39,27,40,35,28)(20,32,34,45,33,46,37,29)(25,41,44,38,42,48,43,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(20,33)(25,42)(29,45)(32,46)(34,37)(38,47)(41,48)(43,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,25)(19,29)(21,37)(22,34)(23,33)(24,38)(26,44)(27,42)(28,41)(30,32)(31,45)(35,43)(36,46)(39,48)(40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135381);